Efficient Phase Noise Modeling of a PLL-Based Frequency Synthesizer

(1)

Proceedings of the 4th International Symposium on Communications,

Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March 2010

Efficient Phase Noise Modeling of a PLL-Based Frequency Synthesizer

Goulven Eynard, Noelle Lewis, Dominique Dallet and Bertrand Le Gal

For small values of PN fluctuations (14)(t)1

«

1rad.) equ. (1) can be simplified to :

Section III presents the new efficient method to generate arbitrary PN spectrum. Finally, Section IV presents the performance evalu- ations of the proposed PN generation architecture. The results are compared to the measured PN spectrum of an existing frequency synthesizer.

III. NOISE GENERATION ARCHITECTURE

A PN noise generator derived straightforwardly from [11] for PLL oscillators is plotted on Fig.2 (a). This architecture requires several independant Gaussian noise generators in order to obtain a satisfying precision. It involves also a +3 dB imprecision error at each frequency corner of the spectrum of the generated PN, due to the superposition of different spectral noise shape [11]. PN generator architectures proposed in [7], [9] are initially not supposed to include additionnal independant Gaussian noise generators, but

II. PHASE NOISE IN A PLL-BASED OSCILLATOR For an ideal oscillator operating at frequencyfo,the voltage output can be expressed asv(t)= A cos(wot+4» where Ais the amplitude,

4>

is a fixed time-invariant phase reference, and Wo = 21rfo is the

carrier pulsation, inrad.s-1.In a practical oscillator, both amplitude and phase are time-varying, which impacts the spectrum of the oscillator by spreading the power of the oscillator around the carrier frequency. In most practical cases, the disturbance in the amplitude is negligible [7] and therefore, only the random deviations of the phase can be considered:

(2) v(t)= Acos(wot

+

4>(t)) (1)

v(t)~A cos(wot) - Ao4>(t ) sin(wot)

The spectrum of

4>(

t) is thus frequency translated around the carrier frequency at ±wo.To quantify this PN, a commonly used spectral measure is the single-sided band (SSB) spectral noise density L(~w) (dBc/Hz):

L(~w)

⁼ ¹⁰¹⁰ (noise power in I.-Hz BW @ Wo

+ ~w)

⁽³⁾

g earner power

where ~w is the carrier pulsation offset from the nominal carrier frequency pulsationWo of the oscillator. A PN spectrum commonly encountered at the output of a PLL-based frequency synthesizer is plotted on Fig.l [12]. The PLL PN is usually majoritarily dominated by the reference crystal oscillator (CO) below the loop bandwith

WBW and by the voltage controlled oscillator (VCO) aboveWBW [5].

The resulting PN spectrum observed at the output of the frequency synthesizer on the frequency interval

[f

min;

f

max] results in a composite spectrum (see Fig.l), composed of three zones of

1/

fa spectral noise shape, delimited by two corner pulsationsWeI andW e2.

In the context of a PLL-based frequency synthesizer,Q is successively equal to 3,0 and 2 [5], [12].

Abstract-This paper proposes a computationaly-efficient procedure dedicated to time-domain phase noise (PN) modeling of a phase- locked loop (PLL)-based frequency synthesizer in a radio-frequency (RF) context. The structure proposed is able to describe any kind of PN including thermal and flicker noise regions using time-domain equations.

The design process of the phase noise modeling process is detailed. An expression of the estimated mean square error (MSE) of the frequency response of the PN generating system is proposed and compared to a previously proposed PN generating system. The spectral behaviour of the simulated frequency synthesizer is compared to practical PN measurements obtained with a spectral analyzer.

I. INTRODUCTION

Phase noise (PN) is a crucial parameter that can limit the overall performance of contemporary wireless [1] and optical [2] transmission systems. Efficient evaluation of the PN impact on the overall performance of complete transmission systems are required. In this context, simple and computationally efficient design methods of system-level PN generators are of particular interest.

Numerous methods exist in literature concerning PN modeling [3- 11]. One of the major difficulty is flicker noise modeling. Such non- stationnary noise can be efficiently approximated from a Gaussian noise generator filtered through a suitably designed multirate filter bank. Practical realizations of flicker noise generators approximate the frequency response of the desired filter from a set of recursively defined one-order filtering cells. Perfect synthesis assumes an infinite number of cells, but several approximations exists in the litterature [6-11] in order to obtain practical systems. In [6] IIR and FIR filtering approximation methods are suggested to generate

1/

fa noise sequences. However, this solution turns out to be computationally complex for wideband colored noise generation, particularly for close-to-carrier noise modeling. In [9] a simplification of this method using multirate filter bank process is proposed, but the design method of such multirate filters leads to non-straightforward design methods. Acomputationally efficient and event-driven PN generation scheme were previously proposed in [7] for a free-running oscillator.

However, this design method is not applied in the particular case of PLL-based oscillators.

PN spectrum shapes of free-running and PLL-based oscillators are very different [5], and thus, the PN generation techniques can differ slightly. The purpose of this paper is to develop a simple and computationally efficient design method of a PLL-based PN generator. The results are asserted through a comparison between practical measurements on a PLL-based frequency synthesizer and the proposed PN generation scheme.

The rest of the paper is organized as follow. In Section II, the general characteristics of PN in PLL-based oscillators are shortly adressed.

Goulven Eynard, Noelle Lewis, Dominique Dallet and Bertrand Le Gal are with the IMS-Bordeaux 1 University - IPB ENSEIRB MATMECA, 351 Cours de la Liberation,33405,Talence Cedex, France; e-mail:goulven.eynardwims-bordeaux.fr, dominique.daIlet@ims-bordeaux.fr, noeIle.lewis@ims-bordeaux.fr, bertrand.legal@ims- bordeaux.fr

(2)

[i\J]

. I •• 1_

I •• 1_

. .

: ..1:"" ..

f- .

I(a) Initial PN generator architecture

I

^Gaussian

L ~t(/\I

^_

noise generator : \ ( ; )

I 1,. .&,

maw

=

^wcl w "'IU"

veo^{influenc e}

Loop bandwidth -,

_...

_~

/

W_{m in}

'--- ~

w(log scale) ...••••.

I

H(jw )

F

(dB)A reference CO influence

Fig. I. Typical PLL oscillator phase noise spectrum.

A.

1/ i"

noise generation on [fmin ;fmax]

The aim is to approximate the transfer function of a filter on a given frequency range

[h

^j

f

H].Basically, in order to choose the frequency band

[h ; f

H] in term of the wanted frequency band generation [fmin ; f max], the following values are choosen [II] :

(7)

(6)

(9)

(to)

W i

=

"(i-l)+1 /4WL

w; =

"(i-l)+3/4WL , N

Happrox(P)

=

Ho

II

Hi(p)

i=l

I(b) Proposed PN generator archite cture I

whereN is the number of elementary cells of the filter and : approximated through a cascade of one-order integration cells [13] :

whereWi (resp.w;) are the poles (resp. zeros) of the approximated transfer function and 1 ::;i ::;N. N is the order of the resulting digital filter and is chosen in order to obtain the optimal trade-off between the precision of the approximation of the functionH(p)and the overall computational complexity of the system. The expression of N in term of the precision of the model (estimated mean square error (MSE)) is an important design parameter which will be quantified further. Once the parametersWL, WH and N are fixed, the design parameter "( is defined :

Fig. 2. Plot of initial (a) and proposed (b) time-domain phase noise generator architectures.

where 1 ::;i ::; N. The generalization to any kind of slope can be obtained by following the principle that the distance between each cell stay the same, whereas the distance between the pole and the

"( = (:: r/

^N ⁽⁸⁾

This parameter defines simultaneously the distance on a logaritmic scale between each elementary cell and also the poles and the zeros position of each elementary cell. The general principle of 1/

f

generating noise is illustrated on Fig.3-(a). The recursively defined values of the poles and zero for this particular case, defined in [II]

is reported here : [h jfH]

=

[fmin /lO; lO·fmax]

(4)

This can be obtained by a transfer function defined as :

they are optionally proposed when a composite and multi-zones PN spectrum shape like the one plotted on Fig.l is required.

The architecture initially proposed requires to adjust the power at the input of each filter, allowing the desired separation of each zone and finally the construction of the desired PN spectrum. Previously proposed design process of PN generating architecture requires an optimization problem resolution to determine the gain at the input of each designed filter [9]. On [7] is proposed an arbitrary overall correction of 5.5 dB or 1.5 dB, depending if the different Gaussian noise sources are correlated or not. All of these techniques require an important design process.

From the time-domain flicker noise generation technique of [II], a generalization is now proposed in this paper in order to generate any composite PN spectrum in a computationally efficient way and with a facilited design process.

The most critical step for discrete-time noise modeling operation is the design of filters for 1/

r

noise, where 0

<

a

<

2. Basically, and as suggested in [II], all cases wherea

>

2 can be derived from the previous cases by using classical one-order integration. This is the strategy proposed in [II] for 1/

r

PN modeling, reported on Fig.2 (a). The following part focuses on the generalization of the previously proposed filter design for any value of a on a desired frequency band [fmin ; f max]. This generalization leads to an easier composite PN generation design model, as it will be shown in the second part.

The precision of the new model is also logically enhanced, as shown in the performance evaluation part. The non-integer order filtering operation is higlighted on Fig.2-(a) and (b) by a grayed-background block.

(1 ⁺

^P

)"/2

H(p)

= n; (

WH) 0< /2' (5)

I+ ^L

WL

where H0 is the overall gain of the filter, W H

=

21r

f

^Hand

WL

=

21r

h.

The transfer function of this a /2 order filter can be

(3)

0.0 020

IHi(jw )l (dB ) A

TI

L - ~

W_i+_{1 W ' H I} (log scale)

(H ) (;) (H i) (H) (il (i +l)

Y W L Y WL Y WL Y WL Y WL Y WL

:(8)1/fphase noise generation

I I

^(b)1/Fphase noise generation

I

Fig. 3. 1/ f phase noise generation principle and extension to 1/ f a noise generation principle illustration.

Estim ated Mea n Square error

0 .0030

,\ ' , ,

. . . .

.\ l

!~.i~!~I.~~.g~."!~~~j!~I1.~~~~!~~c::~~~!li .

0 .00 25 \ \

i / i i

¹ ¹ ¹

...\ . ~ ~ : : ; j .

" , ,

. . .

\ i i

¹ ¹ ¹

000 15

,·,,·,·,··:'·\···,,·:"" Proiiosed·P;.{iiene'raiion'arciiiiecture

i \, i/ !

1 1 1

...~ ~'~ : : ; j .

0 .00 10 : / ' " , : : : :

oooos ~

^>•••

>i:,",.. ~c",.::~:::::l:~:~~:~

0 .0000 1

zero of each cell is modified. Following this principle, the following equations are straightforwardly obtained :

Wi

=

,(i- l )+ ( 1/ 2-a/4) WL

I _ (i - l )+ ( I/ 2+a/4 )

W i - , WL ·

(11) (12)

Fig. 4. Precision and computational complexity of the initial and the proposed PN generation architecture in term of the number of cellsN .

The general principle of

1/ t "

generating noise is illustrated on Fig.3- (b). On the following part and from the proposed generalization, an efficient design method for composite PN spectrum is now proposed .

B. Composite PN generation spectrum

Considering a composite PN spectrum as proposed on Fig.I, the optimal transfer function is now defined as :

(22) (21) (18) (19) (20)

a, = a; =

^c.T;

bi

=

W iTs

+

2

b; =

W iTs - 2 with

with:

Discretization of the filter is carried out using the Tustin transforma-

. 2 z-1 h T ' h l' iod Th

tion : p

=

^T_s ^{z + I '} were s ISt e samp mg pen . us :

N

+

I -1

""' ai a iz

Hd(Z)

=

CO

+

_~ b,

+

b;Z-1 (17)

The resulting time domain equation can be written :

N

y[n]

=

cox[n]

+ L

Ji[n]

i = 1

(13)

(14) (

P )

3/ 2 (

p)

1+ 1+

H(p)

=

Ho ^Wel WH

1+.E.... 1+..E....

WL W c2

The approximation function from Fig.2-(b) can be defined as :

C. Discrete-time noise generation

The obtained analog transfer function has to be decomposed in a sum of one order cells in order to decrease the overall numerical error in the computation of the filter coefficients. The decomposed version of the filter equ. (6) is of the form :

~ 1+ ...E..-~ 3/2

where I1~1H}b)(p)approximate ~ ^.^Itcan be predicted that for the same number of cells ,th;L SE of the architecture of Fig.2-(b) is reduced compared to the architecture of Fig.2-(a), since the interval of approximation is[Jmin ; fel], compared to the interval [fmin ; fma x] for he same values of cells N .This results is asserted through simulations .

J(N) =

~E IIH(JW(k»)I- fJ.IIH!~;,t;!ox(JW(k»)W

(23)

where w(k) is in a discrete interval !21l"h;21l"fH] composed of M logarithmically-spaced samples,

H

a~~rox(jw(k)) (resp.

H~~prox(jw(k)) is the transfer function of architecture plotted on Fig.2-(a) (resp. Fig.2-(b» . As an illustrative example , the measured spectrum of an Agilent 8662A using a spectrum analyzer using [i.

=

100Hz, fH

=

1MHZ is plotted on Fig.6. The parameter of the desired frequency response(Jel

=

3kH z ,[ ez

=

1MH z ) are The design of the proposed PN generator leads to a similar architecture as the one proposed in [7]. Similarly, a multi-rate process and anti-aliasing filter can be efficiently used. In the next part, the performance of the current noise generation process is compared to the original process proposed in[11].

IV. PERFORMANCE EVALUATION

A. Frequency response MSE

An estimation of the MSE frequency response J(N) between the ideal filter and its practical implantation in term of [t., fH and N can be written:

(16) (15) N

H

approx

^{( a ,b)} ^{( ) _}^{p -} ^CO

⁺

" " '^L.J ~+c,

i = 1 ' P with

(4)

I I " '" ' ' ' " ., , , , , ' " " , , " ' " " ' "

I I " '" ' ' ' " I " , . , . . , ' " " . . , ' " " ~ '" ~

····,-- ' · T·.· , · · · - ···· ..· · ..·.· ..' " r · · · ' - ,- , , - ..-,- .

jll~·II·IIII,···I··I·IIIILI;I;III"I···I·IIII,

....:..

~ . ; .:.,~,:;

...

'~"" '-'k;'H·,r:'ri ' ·'^r;;~ ji ;~· --~I}"II\ ' . --.:. . ;. ;.:., ,,:

i iiiiiiii r~~:'~r"~!';'1r"rr : : : ^:1:: ,!~; 1~ ~ ~ T

I I " ' " ' ' ' " " '"I I I I" " " ' " . , , , , " " ~'" ~

----,--,

...

-.-,...,----

..

^{- -}^...^-.. .. .. ...----,--,.. -.-.... ..."...---,--..-.. ...-,-...., ----. ,

....

^,

^{... ...}

I I " '" ' ' ' " " " " I " " " " ' " " ' " " " " ' "

I I " '" ' ' ' " ,.. . " ' " .,, , , , , . ," . . , , " '' ' '

I I I " " " I I I" " " ' " . , , , , , " , , , .

, " " ' ''' , " " " " " " ' ' ' ' , "".

I I " ' " ' ' ' " ,.". . I I I" " " ' " '' ' " " " ~ '" ~

I I " '" ' ' ' " ., , , . . ' " " " " ' " " , " " " '

---.:.-~-

..

^: ^,.:^..^{---- - -}

..

-~

..

^{~ ,. -}^--

-.

---,- ---..--~-~

..

-,-,.... ----,--..

-.

...,

, " " "" ' " " " " ' " " " " ' " " ' " " "

, " "

..

^" ^' ^" " " " ' " " ,

..

^, ^{' "} ^"

..

^, ^" ^{' "}

, " " " " ' " " " " ' " " " " ' " " '" ' " ""

, " "

..

^" , , ' """ ' " ",

..

^, ^{' "} "," ' " "",

, " "

..

" ' " " " " ' " "

..

" ' " " '" ' " " " '

, " " ' " ' ' " " " " ' " " ,

..

^, ^{' "} ^"

..

^, ' " " " '

\ i !i !i !ii! i! ii!ii!i i i!ii !!!i i iiii!!!: !i! i!i!i

T]~'tw Tn'i'[~ jIIl']],, ]n-mf UiI[';

_", ' "_{" ~}_{' '' '}"" " ' " '' ''' " ' " ' " "' "' " ''''" " " "' " ' " ' ' ''' ' ' ''' "' "' '' '' '" '"

----I·+I·I·r~:i,\~·~·HI,lr~----1--1·.-;1M\H:lj-l-li;·----H·,·I·-I-I

--.. i""1·l·,ri·il=~~,~Wl\~~~~~::lii¥·~i~w;;~;iiF,~·H+rfi

, " " " " " " ' " ' " ' " " " " " " " "

- - - - ,- - ., -~ - ,- t" ~ t",.. - - - -t" . ... - ... - r~ ~ r... - - - -~ --t"-, -~... ..~ t"- - - . , -- r - r ...• , ~-, - t" rt",

, " " " " ' " ' " 11' ' " " " " " " .. " " "

, " " " " ' " , '" " , ' " "" " ' " " ' " " ' "

, " " " " ' " "

..

" ' " " " " ' " ,,, .. " ' "

, " " II " ' " " ' II ' ' " " " " " " " .. , "

, " " " " ' " ' " ' " ' " " " " ' " ' " '' ' "

, " " "" ' " , " , " , , ' " " " ' " ' ' ' ' ' ' II

, " " "" ' " ' " '" ' " " " " ' " ,,, .. , ,

_.--,--..-^~-,-,.~,. ,..---- ,.

...

-

..

-^~

.

^{~ ~ ,.-}---^{~ .}-,.-,-

... ....

^{~ ,.}---..--^~-^~

..

-,-,....----,--..

, " " " " ' " ' " 11' ' " " " " ' " ,,,.. "

, " " " " ' " ' " ' " ' " " " " ' " ' " '' ' "

, , , ', ''' ' " " " 11' , , ' " " " " , ' " ' ' ' "

, " " " " ' " ' " ' " ' " " " " ' " ' " '' ' "

, " " "" ' " ' " 11' ' " " " " ' " , , , " " "

, " " " " ' " ' " 11' ' " " " " ' " ,,, .. " ' "

, " " "" ' " ' " 11' ' " " " " ' " ' " '' " ' "

·8 0

·9 0

· 10 0

· 110

~

CO · 12 0

~ 0-rJl

0 · 13 0

· 140

· 15 0

· 16 0 10

s

10 10

1 '2

10

1 l HzI

·90

· 10 0

· 1 10

~

CO ^{· 120}

~0- · 130

rJl 0

· 140

·1 50

· 16 0

10 10 1" ¹⁰ ¹⁰ 10/,2 ¹⁰ ^{fiHz )}

(24) Fig. 5. Agilent 8662A frequency synthesizer measured phase noise spectrum (/0= 3.10⁹Hz) .

extracted from theses measurements. The MSE obtained for various values of cells N are reported , assuming,

is

⁼ ^{io /100}⁼ 30MHz, and M

=

800. The results compared to the initial architecture proposed on [11] are compared on FigA .

From this figure, it can be noted that the estimation precision of the architecture plotted on Fig.2-(a) is biased . Below a certain value ofN, adding new cells will not increase the precision of the filter, while still increasing linearly the computational complexity of the system . This bias can be partly explained by the +3 dB error at each frequency comer. On the other part, the obtained MSE of the proposed architecture is lower and exhibits no bias.

B. Experimental Results

From the same measured PN spectrum (see Fig.5), using the previously designe filter response, the frequency results are analyzed in the frequency-domain with a periodogram. The periodogram performs an average on several sub-windows of length M

=

8192 , distributed over the discrete-time signal obtained at the output, with an over- lapping between each consecutive window of 50%. Considering the effect of a rectangular windowing on the periodogram, the theoretical and the measured PSD are related by :

psd meas

=

PSdtheo

+

10log (Ts2M )

The spectrum of the noise obtained is plotted on Fig.6. The asymp- totic values are plotted on Fig.5 and Fig.6 in order to make an efficient comparison. The simulated PN well matches the measured results .

V. CONCLUSION

On this paper, a computationally-efficient time-domain PN generation model has been proposed. A generalization of [II] design technique was given and from this technique , an efficient time domain PN generation architecture was proposed . The design process of the new proposed PN modeling architecture was clearly explicited.

The results are asserted through evaluation of the estimated MSE of the frequency response of the proposed filter, compared to the previously proposed architecture. Finally, a comparison between the

Fig. 6. Power spectral density of the generated phase noise spectrum .

PN measurements of a RF frequency synthesizer and the generated PN was proposed to validate the procedure.

ACKNOWLEDGEMENT

The work is supported by the DGA - SCERNE project.

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r

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Efficient Phase Noise Modeling of a PLL-Based Frequency Synthesizer